OpenAI model disproves Erdős unit distance conjecture

For decades the working consensus was that square grids — Erdős's own construction — were essentially optimal, yielding roughly n^(1+c/log log n) unit-distance pairs and motivating the conjectured upper bound of n^(1+o(1)) ^[2]. The AI proof breaks that ceiling by abandoning fixed-degree settings like Gaussian integers and instead taking points in a CM number field K whose degree grows with the size of the configuration, building on ideas from Ellenberg–Venkatesh and Hajir–Maire–Ramakrishna ^[2]. The key ingredient is an infinite class field tower of Golod–Shafarevich type: a sequence of progressively larger unramified extensions whose existence is guaranteed by Golod–Shafarevich-style inequalities and which supplies enough algebraic units to force many distinct pairs of points to lie at distance exactly one. Will Sawin later refined the argument by reducing the required input to a single rational prime splitting in the tower ^[2]. The result is a sequence of point sets in R^2 with at least |P|^(1+ε) unit distances for some fixed ε > 0 — superlinear by a polynomial factor, not a logarithmic one — establishing the first known violation of the n^(1+o(1)) belief ^[3].

The model was not a math-tuned scaffold like a theorem-prover or a symbolic search system — it was an internal general-purpose reasoning model, and the proof was 'first mathematically generated in one shot' before being expositionally refined through human interactions with Codex ^[2]. That distinction matters: prior AI math results have generally either solved 'amusement-grade' Erdős problems or required heavy domain scaffolding. Here, a single reasoning model produced an attack route that draws on infinite class field towers, Golod–Shafarevich inequalities, and CM number fields — tooling that lives in algebraic number theory, not discrete geometry ^[2]. Reported telemetry puts the cost at 5–32 hours and roughly $120–$1,000 in tokens, producing a 125-page chain-of-thought summary whose crucial insight surfaces on page 39 ^[4]. As Jacob Tsimerman put it, AI systems have an edge because they can 'play for longer and in more treacherous waters than mathematicians without getting overwhelmed' ^[2]— a description of stamina-plus-recall, not of a specialized solver.

Melanie Wood's verification note is unusually pointed: 'if the level and type of human expertise that is represented on this note had been assembled to find a counterexample to this conjecture a month ago, the mathematicians would have found a counterexample. However, without the claimed proof by ChatGPT, there is no particular reason anyone would have tried to look for a counterexample' ^[2]. Two anchoring forces produced that gap. First, Erdős himself consistently believed the upper bound n^(1+o(1)) was tight, and he was clear enough about it that, as Bloom notes, either a proof or disproof of that bound counts as winning his $500 prize ^[2]. Second, the toolkit required — infinite class field towers and Golod–Shafarevich constructions — sits firmly in algebraic number theory, while the unit distance problem is canonically a combinatorial geometry question; few researchers carry both halves of that intersection. The AI did not respect that disciplinary boundary, and as Arul Shankar put it, 'current AI models go beyond just helpers to human mathematicians — they are capable of having original ingenious ideas, and then carrying them out to fruition' ^[2].

The result is a disproof by construction, not a solution to the full unit distance problem. The best known upper bound is still Spencer, Szemerédi, and Trotter's O(n^(4/3)) from 1984, and the gap between that bound and the new lower bound of n^(1+ε) is enormous ^[2]. Bloom's own framing — that the result reveals number-theoretic depth but introduces no new geometric tools — is echoed across the mathematician community: this is an existence proof that the old belief was wrong, not a new structural understanding of unit-distance graphs ^[1]. The Remarks paper also flags methodological wrinkles. The AI-generated text did not cite Hajir–Maire–Ramakrishna, the prior work most directly underlying the construction, raising attribution norms that the field has not yet codified ^[2]. Wood's broader concern is downstream: if AI models begin producing long, technical-looking proofs at scale, the risk is not lack of verification on a celebrated problem like this one, but a flood of plausible-but-wrong manuscripts in less glamorous corners of mathematics where nine senior co-verifiers will not show up ^[1].